1. Implementation of Public Key Encryption Algorithms
CSCI 5857: Encoding and Encryption

2. Outline Fast modular exponentiation Finding very large prime numbers
Rearranging multiplication and mod
The square and multiply method
Finding very large prime numbers
Generate and test method
The Miller-Rabin test

3. Modular Exponentiation
Must compute C = P E mod n and P = C D mod n for very large n, E, D, P, and C
Example:
Problem: P E is a really big number!
Too large for fast computation or even storage on most systems
Note: Final result P E mod n ≤ n

4. Modular Multiplication Properties
Solution: Keep value smaller by taking mod n throughout computation (instead of at end)
Useful property of modular arithmetic: a  b mod n = ((a mod n)  (b mod n)) mod n
Example: mod 17 = (37  58) mod = (37 mod 17)(58 mod 17) mod = 3  7 mod 17 = 4

5. Square and Multiply Method
Break exponent E into product of powers of 2
Example: P22 = P16  P4  P2
Can be represented as “bits”: 22  10110
Use squaring to compute powers of 2 quickly
P  P 2  P 4  P 8  P 16
Multiply by running total if corresponding bit = 1

6. Fast Modular Exponentiation
Algorithm to compute PE mod n: result = 1 for (i = 0 to number of bits in E - 1) { if (ith bit == 1) result = (result * P) mod n P = P2 mod n }

7. Fast Modular Exponentiation Example
Example: 1722 mod 21 i
ith bit P
result 17
(unchanged) 1
172 mod 21 = 16
1 * 16 mod 21 = 16 2
174 mod 21 = 162 mod 21 = 4
16 * 4 mod 21 = 1 3
178 mod 21 = 42 mod 21 = 16 4
1716 mod 21 = 162 mod 21 = 4
1 * 4 mod 21 = 4

8. Finding Large Primes
Need very large primes (hundreds of digits) for key generation
Problem: No simple way to generate an arbitrarily large prime number
Usual method: generate and test
Generate a sufficiently large odd number
Test it for primality

9. Frequency of Primes
How many might we need to test before finding a prime?
Primes of size s are spaced about every ln(s)
ln(s) = natural logarithm = loge(s)
Example: Finding prime of size 2200 would require testing on average ln(2200)  138 numbers

10. Miller-Rabin Test
Basic idea: many primes are of form p = 2kq + 1 for some k, q
Miller-Rabin test: If p is prime then for all a < p – 1 either:
a q mod p = 1
One of a q, a 2q, a 4q, … a 2k-1q mod p = -1

11. Probabilistic Testing
Problem: No way to test all possible values of a
2200 possible values of a for primes of size 2200
Probabilistic approach: Try enough values of a to be as sure as necessary that p is prime
Odds that non-prime p passes test for some a: 1/4
Odds that non-prime p passes t different tests: (1/4)t
p passes15 tests  odds are 1 in a billion that it is not prime

12. Algorithm for Finding Large Primes
Possible algorithm:
Generate sufficiently large p = 2kq + 1 for some k, q
Test for divisibility by small primes (3, 5, 7, 11, 13, 17)
Quick elimination of most failed candidates
Choose a set of test a’s for Miller-Rabin test
Enough to be as sure as necessary
Example: 15 for % certainty
If p passes test for all a’s, p can be used as prime

13. What’s Next Let me know if you have any questions
Continue on to the next lecture on Attacks on Public Key Encryption